Problem: What is the average rate of change of the function $g(x)=\sqrt{x}$ over the interval $5\leq x \leq t$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\sqrt{t-5}}{t-5}$ (Choice B) B $\dfrac{\sqrt{t}-\sqrt{5}}{5}$ (Choice C) C $\dfrac{\sqrt{t}-\sqrt{5}}{t-5}$ (Choice D) D $\dfrac{\sqrt{t-5}}{t}$
Solution: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We are interested in the average rate of change of $g(x)=\sqrt{x}$ over the interval $5\leq x \leq t$ : $\begin{aligned} &\phantom{=}\dfrac{g(t)-g(5)}{(t)-(5)} \\\\ &=\dfrac{\sqrt{t}-\sqrt{5}}{t-5} \end{aligned}$ The average rate of change of the function is $\dfrac{\sqrt{t}-\sqrt{5}}{t-5}$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints.